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On Asymptotics of Exchangeable Coalescents with Multiple Collisions

Part of: Special processes Combinatorial probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander Gnedin ,
Alex Iksanov  and
Martin Möhle
Alexander Gnedin*
Affiliation:
Utrecht University
Alex Iksanov*
Affiliation:
National Taras Shevchenko University of Kiev
Martin Möhle*
Affiliation:
University of Düsseldorf
*
∗Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. Email address: a.v.gnedin@uu.nl
∗∗Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kiev, 01033 Kiev, Ukraine. Email address: iksan@unicyb.kiev.ua
∗∗∗Postal address: Mathematical Institute, University of Düsseldorf, Universitätsstrasse 1, 40225 Düsseldorf, Germany. Email address: moehle@math.uni-duesseldorf.de
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Abstract

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We study the number of collisions, Xn, of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite characteristic measure η(dx) = x−2Λ(dx). Via a coupling technique, we derive limiting laws of Xn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of Xn include normal, stable with index 1 ≤ α < 2, and Mittag-Leffler distributions. The results apply, in particular, to the case when η is a beta(a − 2, b) distribution with parameters a > 2 and b > 0. The approach taken allows us to derive asymptotics of three other functionals of the coalescent: the absorption time, the length of an external branch chosen at random from the n external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.

Keywords

Absorption timecouplingexchangeable coalescentexternal branch lengthmultiple collisionsnumber of collisionsregenerative composition

MSC classification

Secondary: 60G09: Exchangeability 60C05: Combinatorial probability 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1186 - 1195
Copyright
Copyright © Applied Probability Trust 2008 

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